], we proposed a deterministic method to design an ultrahigh Q PhC nanobeam cavity and verified our designs experimentally. The proposed method does not rely on any trial-and-error based parameter search and does not require any hole shifting, re-sizing and overall cavity re-scaling. The key design rules we proposed, that result in ultrahigh Q cavities, are (i) zero cavity length (L = 0), (ii) constant length of each mirror segment (‘period’=a) and (iii) a Gaussian-like field attenuation profile, provided by linear increase in the mirror strength.

In this follow-up work, we provide numerical proof of the proposed principles, and systematically optimize the design recipe to realize a radiation limited cavity and waveguide coupled cavity. Furthermore, we extend the recipe to the design of air-mode cavities, whose optical energies are concentrated in the low-index region of the structure.

Nanobeam cavities have recently emerged as a powerful alternative to the slab-based 2-D PhC cavities [17

]. We note that the same design principle discussed here could be directly applied to realize ultra-high Q cavities based on dielectric stacks that are of interest for realization of vertical-cavity surface emitting lasers (VCSELs) and sharp filters. Finally, it is important to emphasize that while our method is based on the framework of Fourier space analysis [35

], alternative approach, based on phase-matching between different mirror segments, could also be used to guide the design, as well as to explain the origin of deterministic ultra-high Q-factors in our devices [38

]. It consists of an array of air-holes in decreasing radii, etched into a ridge waveguide. The hole-to-hole distances (“periodicity”) are constant. The structure is symmetric with respect to the dashed line in Fig. 1(a). In contrast to the majority of other cavity designs, current structure has no additional cavity length inserted between the two mirrors (L=0), that is the hole-to-hole spacing between the two central holes is the same as the rest of the structure (a). This minimizes the cavity loss and the mode volume simultaneously. The cavity loss is composed of the radiation loss into the free space (characterized by Qrad) and the coupling loss to the feeding waveguide (Qwg). Qwg can be increased simply by adding more gratings along the waveguide. Qrad can be increased by minimizing the spatial Fourier harmonics of the cavity mode inside the lightcone, achieved by creating a Gaussian-like attenuation profile [32

]. The optical energy is concentrated in the dielectric region in the middle of the cavity (Fig. 1(b)). In order to achieve the Gaussian-like attenuation, we proposed to use a linearly increasing mirror strength along the waveguide [32

Fig. 1 (a) Schematic of the proposed nanobeam cavity. (b) FDTD simulation of the energy density distribution in the middle plane of the nanobeam cavity.

First, we analyze the ideal tapering profile using plane wave expansion method and verify the results with 3D FDTD simulations. The dielectric profile of the structure in the middle plane of the cavity can be expressed as

1ε(ρ)=1εSi+(1εair−1εSi)S(ρ)

(1)

with

S(ρ)={1|ρ−rj|≤R0|ρ−rj|>R

rj = j · ax̂, a is the period, and j = ±1, ±2... are integers. R is the radius of the hole. Using plain wave expansion method [40

J1 is the first order Bessel function. Filling fraction f = πR2/ab is the ratio of the area of the air-hole to the area of the unit cell. We note that the above expressions are calculated assuming nanobeam cavity has infinite thickness (i.e 2D equivalent case). Better estimation can be obtained by replacing εair and εSi with the effective permittivities.

The dispersion relation can be obtained by solving the master equation [41

Inside the bandgap, the wavevector
(k) for a given frequency (ω) is a complex
number, whose imaginary part denotes the mirror strength
(γ). For solutions near the band-edge, of interest for
high-Q cavity design [32

], the frequency can be written as ω=(1−δ)κ0πc/a (δ is the detuning from
the mid-gap frequency) and the wavevector as k = (1
+ iγ)π/a.
Substituting this into the master equation, we obtain δ2+γ2=κ12/4κ02. The cavity resonance asymptotes to the dielectric
band-edge of the center mirror segment: wres→(1−κ1j=1/2κ0j=1)κ0j=1πc/a (j represents the
jth mirror segment counted from the center), at
which point the mirror strength
γj=1 = 0.
γ increases with j. With
εair = 1 and
εSi = 3.462, we calculate
in Fig. 2(a) the γ
– j relation for different tapering profiles. It can be
seen that quadratic tapering profile results in linearly increasing mirror
strengths, needed for Gaussian field attenuation [32

Fig. 2 (a) Mirror strengths of each mirror segment for different tapering profiles obtained from the plane wave expansion method (‘1’ indicates the mirror segment in the center of the cavity). (b) Band diagram of the TE-like mode for f = 0.2 and f = 0.1. The green line indicates the light line. The circle indicates the target cavity resonant frequency. (c) Mirror strengths for different filling fractions, obtained using 3D band diagram simulation. (d) Mirror strengths as a function of mirror number after quadratic tapering. (e) Radiation-Q factors for nanobeam cavities with different cavity lengths between the two Gaussian mirrors, obtained using 3D FDTD simulations. (f) Resonances of the cavities that have different total number of mirror pair segments in the Gaussian mirror, and their deviations from the dielectric band-edge of the central mirror segment, obtained using both FDTD simulation and perturbation theory. (g) Hz field distribution on the surface right above the cavity, obtained from 3D FDTD simulation. The structure has dimension of a = 0.33μm, b = 0.7μm, the first 20 mirror segments (counted from the center) have fs varying from 0.2 to 0.1, followed by 10 additional mirror segments with f = 0.1. (h) Hz field distribution on the surface right above the cavity, obtained from the analytical formula
Hz=sin(πax)exp(−σx2)exp(−ξy2), with a = 0.33μm, σ = 0.14, ξ = 14. (i) Hz field distribution along the dashed line in (g)&(h). Length unit in (g)–(i) is μm.

Next, with the optimized tapering profile, the cavity is formed by putting two such mirrors back to back, leaving a cavity length L in between (Fig. 1(a)). Figure 2(e) shows the simulated Q-factors for various Ls. Highest Qrad is achieved at zero cavity length (L=0), which supports the prediction in [32

Third, we verify that the cavity mode has a Gaussian-like attenuation profile. Figure 2(g) shows the Hz-field distribution in the plane right above the cavity, obtained from 3D FDTD simulation. As shown in Fig. 2(h), this field distribution can be ideally fitted with Hz = sin(πx/a)exp(−σx2)exp(−ξy2), with a = 0.33, σ = 0.14 and ξ = 14. The fitted value a agrees with the “period”, and σ agrees with that extracted value from Fig. 2(d):
σ=dγdxπa=0.13. Figure 2(i) shows Hz distribution along the dashed line in Figs. 2(g) and 2(h). Therefore, we conclude that zero cavity length, fixed periodicity and a quadratic tapering of the filling fraction results in a Gaussian field profile, which leads to a high-Q cavity [32

],
current method results in a cavity whose resonance is asymptotically approaching the
dielectric band-edge frequency of the central mirror segment (circled in Fig. 2(b)). The deviation from the band-edge
frequency can be calculated using perturbation theory [41

E|| is the component of
E that is parallel to the side wall surfaces of the holes and
D⊥ is the component of D that is
perpendicular to the side wall surfaces of the holes. Under Gaussian distribution,
the major field component Dy =
cos(π/ax)exp(−σx2)exp(−ξy2),
δε perturbation occurs at r
= ±(j − 1/2)a +
Rj, where Rj=fjab/π denotes the radius of the
jth hole (counted the center), with
j=2,3...N, N is the
total number of mirror segments at each side. Since the cavity mode has a Gaussian
profile, 1/σ characterizes the effective length of the cavity
mode, and scales linearly with N, with a nonzero intercept due to
diffraction limit. For large N, the intercept can be neglected, and
thus σN=20×0.142/N. Plug the perturbation induced by the quadratic
tapering from f = 0.2 to f = 0.1
into Eq. (6), the frequency offset
δλ/λ v.s
N can be obtained. Figure
2(f) shows the frequency offset for different total number of mirror
pairs (N), calculated from the perturbation theory, as well as
using FDTD simulations. It can be seen that the deviation decreases as the number of
modulated mirror segments increases, and is below 1% for N
> 15.

Therefore, we verify that an ultrahigh-Q, dielectric-mode cavity resonant at a target frequency can be designed using the following algorithm:

Determine a target frequency. For example in our case we want ftarget = 200THz. Since the cavity resonant frequency is typically 1% smaller than the dielectric band-edge of the central segment, estimated using the perturbation theory, we shift-up the target frequency by 1%, i.e. fadjusted = 202THz.

Pick the thickness of the nanobeam - this is often pre-determined by the choice of the wafer. For example, in our case, the thickness of the nanobeam is 220nm, determined by the thickness of the device layer of our silicon-on-insulator (SOI) wafer.

Choose periodicity according to a =
λ0/2neff,
where neff is effective mode index of the
cavity and can be estimated by numerical modeling of a strip waveguide
that nanobeam cavity is based on. However, we found that the absolute
value of the periodicity is not crucial in our design, as long as there
exists a bandgap. Therefore, we pick neff
= 2.23, which is a median value of possible effective indices in
the case of free standing silicon nanobeam
(neff ∈ (1, 3.46)). This results
in a = 330nm.

Set nanobeam width. Large width increases the effective index of the cavity mode, pulls the mode
away from the light line, thus reducing the in-plane radiation loss. On
the other hand, a large beam width will allow for higher order modes
with the same symmetry as the fundamental mode of interest. Using band
diagram simulations, we found that the width of 700nm is good trade-off
between these two conditions (Fig.
2(b)).

Set the filling fraction of the first mirror section such that its dielectric band-edge is at the adjusted frequency: 202THz in our case. Band diagram calculations based on unit cells are sufficient for this analysis. We found that an optimal filling fraction in our case is fstart = 0.2 (Fig. 2(b)).

Find the filling fraction that produces the maximum mirror strength for the target frequency. This involves calculating the mirror strength for several filling fractions (Fig. 2(c)), each of which takes one or two minutes on a laptop computer. In our case we found that fend = 0.1.

Pick the number of mirror segments (N) to construct the Gaussian mirror: we found that N ≥ 15 (on each side) are generally good to achieve high radiation-Qs.

Create the Gaussian mirror by tapering the filling fractions quadratically from fstart (=0.2 in our case) to fend (=0.1) over the period of N segments. From the above analysis, the mirror strengths can be linearized through quadric tapering (Fig. 2(d)).

Finally, the cavity is formed by putting two Gaussian mirrors back to back, with no additional cavity length in between (L = 0). To achieve a radiation-limited cavity (Qwg >> Qrad), 10 additional mirrors with the maximum mirror strength are placed on both ends of the Gaussian mirror. We will show in the next section, no additional mirrors are needed to achieve a waveguide-coupled cavity (Qrad >> Qwg).

] (Fig. 1), the alternative
structure which has the air-hole in the symmetry plane, as shown in Fig. 3(a), also satisfies (i)–(ix). Both
structures result in dielectric-mode cavities, since the bandgap of each mirror
segments red-shifts away from the center of the cavity, and thus a potential well is
created for the dielectric band-edge mode of the central segment. The difference is
that the energy maximum in the air-hole centered cavity is no longer located in the
middle of the structure, but instead in the dielectric region next to the central
hole (Fig. 3(b)). Figure 3(c) shows the Hz field
profile in the plane right above the cavity, obtained from FDTD simulation. Figure 3(d) shows the fitted field profile
using the same parameters that are used in the original structure shown in Figs. 2(g)–2(i), but with sine function
replaced by cosine function. Figure 3(e)
shows the Hz distribution along the dashed line in Figs. 3(c) and 3(d).

Fig. 3 (a) Schematic of the Gaussian nanobeam cavity, with an air hole in the symmetry plane (dashed line). (b) Energy distribution in the middle plane of the cavity obtained from 3D FDTD simulation. (c)&(d) Hz field distribution on the surface right above the cavity: (c) is obtained from 3D FDTD simulation and (d) is obtained from the analytical formula
Hz=cos(πax)exp(−σx2)exp(−ξy2), with a = 0.33μm, σ = 0.14, ξ = 14. (e) Hz field distribution along the dashed line in (c)&(d). Length unit: μm.

Armed with the analytical field profile of the cavities: Hzodd(x)=sin(πx/a)exp(−σx2) (Fig. 2) and Hzeven(x)=cos(πx/a)exp(−σx2) (Fig. 3), we
can obtain the radiation losses and far fields of the cavities using the Fourier
space analysis [35

]. The
Fourier transforms can be analytically obtained FT(Hzodd)=(exp(−(k+π/a)2/4σ)−exp(−(k−π/a)2/4σ))/i8σ and FT(Hzeven)=(exp(−(k+π/a)2/4σ)+exp(−(k−π/a)2/4σ))/8σ. Under σa2
<< 1, both distributions have their Fourier components strongly
localized at k =
±π/a, as is verified by FDTD
simulations in Fig. 4(a) and 4(b). Since Hzodd(x) is an odd function, it always has a zero Fourier
component at k = 0. Therefore, dielectric-centered cavities
(Fig. 1) should have higher
Q-factors. However, in high-Q cavity designs,
σa2 << 1 is satisfied and
thus both dielectric-centered and air-centered cavities have comparable
Q-factors. FDTD simulation shows that the above Hzodd and Hzeven cavities have Qtot
= 3.8 × 108 and Qtot
= 3.5 × 108 respectively. The mode volume of the Hzodd cavity is
0.67(λres/nSi)3,
smaller than the Hzeven cavity (V =
0.76(λres/nSi)3).

Fig. 4 (a)&(b) The distribution of the spatial Fourier components of the cavity mode, obtained from 3D FDTD simulation: (a) for the
Hzodd cavity and (b) for the
Hzeven cavity respectively. (c)&(d) The far field profile of the cavity mode obtained from 3D FDTD simulation: (c) for the
Hzodd cavity and (d) for the
Hzeven cavity respectively. The inset cavity structure shows the orientation of the waveguide direction in (c)&(d). Dashed line indicates the symmetry plane.

The far field radiation patterns (obtained using FDTD simulations) of the two cavities are shown in Figs. 4(c) and 4(d). The powers, in both cases, are radiated at shallow angles (> 70° zenith angle) to the direction of the waveguide. The Hodd cavity has even less radiated power at small zenith angles, consistent with the above analysis. By integrating the zenith and azimuth angle dependent far field emission, we found that 32% and 63% of the power emitted to +ẑ direction can be collected by a NA=0.95 lens, respectively for Hodd cavity and Heven cavity.

3. Ultra-high Q, dielectric-mode photonic crystal nanobeam cavities

3.1. Radiation-Q limited and waveguide-coupled cavities

Since the dielectric-centered Hzodd cavity has smaller V than the Hzeven one, we focus our discussion in the Hzodd case. Using the above design algorithm, we
design the Gaussian mirror and put 10 additional mirrors with the maximum mirror
strength on both ends of the Gaussian mirror to obtain the radiation-limited
cavity (Qwg >>
Qtot). We find in Fig. 5 that Qtot increases
exponentially and V increases linearly as the total number of
mirror pairs in the Gaussian mirror (N) increases. A record
ultra-high Q of 5.0 × 109 is achieved while
maintaining the small mode volume of 0.9 ×
(λres/nSi)3
at N = 30.

Fig. 5 (a) Total Q-factors (log(10) scale) and effective mode volumes (V/(λres/nSi)3) of nanobeam cavities for different total number of mirror pair segments in the Gaussian mirror. In each case, 10 additional mirror segments with f=0.1 (maximum mirror strength) are added on both ends of the Gaussian mirror. Therefore, the total-Q of the cavity is limited by radiation-Q. A record ultra-high Q of 5.0 × 109 is achieved with a Gaussian mirror that comprises 30 mirror segments and an additional 10 mirror pairs on both ends. (b) On-resonance transmissions and total Q-factors (log(10) scale) v.s the total number of mirror pair segments in the Gaussian mirror. In this case additional mirror pairs (10 of them) are not included. A record high-T (97%) and high-Q (1.3 × 107) cavity is achieved at N = 25.

Our design strategy has an additional important advantage over other types of photonic crystal
cavities [17

], that is: the cavity naturally
couples to the feeding waveguide, as the hole radii decrease away from the
center of the cavity. High-Q and high transmissions
(T) cavities are possible with the above design steps
(i)–(ix), with no additional “coupling sections” needed.
We study T and Qtotal dependence on
the total number of mirror pair segments in the Gaussian mirror
(N) in Fig. 5(b).
Partial Q-factors (Qrad,
Qwg) are obtained from FDTD simulations, and
T is obtained using T=Qtotal2/Qwg2 [41

3.2. Higher order modes of the dielectric-mode cavity

The ultra-high Q mode that we deterministically designed is the fundamental mode of the cavity. Meanwhile, higher order cavity modes also exist. The number of higher order modes depends on the width of the photonic band gap and total number of mirror segments in the Gaussian mirror. To reduce the simulation time, we study the higher order modes of a waveguide-coupled cavity, that has a total number of 12 mirror pair segments, possessing a moderate Q-factor. Figure 6(a) shows the transmission spectrum obtained from FDTD simulation, by exciting the input waveguide with a waveguide mode, and monitoring the transmission through the cavity at the output waveguide. The band-edge modes are observed at wavelengths longer than 1.6μm and shorter than 1.3μm. Figures 6(b)–6(d) shows the major field-component (Ey) distribution of the three cavity modes. As expected, the eigenmodes alternate between symmetric and anti-symmetric modes. Symmetry plane is defined perpendicular to the beam direction, in the middle of the cavity (dashed line in Fig. 6). The total Q-factors of modes I–III are 10,210, 1,077 and 286 respectively. Effective mode volumes of them are 0.55, 0.85 and 1.06 respectively. We note that transversely odd modes are well separated from the transversely symmetric cavity modes, hence were not considered in Fig. 6.

Fig. 6 (a) Transmission spectrum of the cavity from FDTD simulation. (b)–(d) The Ey field distribution in the middle plain of the nanobeam cavity. Resonances and symmetries of the modes are indicated in the plot. Symmetry plane is indicated by the dashed line. Length unit in (b)–(d) is μm.

4. Ultra-high Q, air-mode photonic crystal nanobeam cavities

4.1. Radiation-Q limited cavity

An air-mode cavity concentrates the optical energy in the low index region of the cavity.
Therefore, these cavities are of interest for applications where strong
interactions between light and material placed in the low index region of the
cavity is required, including nonlinear optics [4

]. The ultra-high
Q air-mode nanobeam cavity is realized by pulling the
air-band mode of photonic crystal into its bandgap, which can also be designed
using the same design principles that we developed for dielectric-mode cavities.
In contrast to the dielectric-mode case, the resonant frequency of the air-mode
cavity is determined by the air band-edge frequency of the central mirror
segment. Then, to create the Gaussian confinement, the bandgaps of the mirror
segments should shift to higher frequencies as their distances from the center
of the cavity increase. This can be achieved by progressively increasing the
filling fractions of the mirror segments away from the center of the structure
(instead of decreasing in the dielectric-mode cavity case). One way to
accomplish this is to increase the size of the holes away from the center of the
cavity. While this may be suitable for non-waveguide coupled
(radiation-Q limited) cavities, it is not ideal for a
waveguide-coupled cavity. For this reason, we employ the design that relies on
tapering of the waveguide width instead of the hole size. Similar geometry was
recently proposed by Ahn et. al. [44

The same design steps can be followed as in the dielectric-mode cavity case, with the following changes: First, the adjusted frequency (198THz) is 1% lower than the target frequency (200THz). (The thickness of the nanobeam is 220nm and period is 330nm, same as previous case.) Second, the nanobeam width at the center of the cavity is wstart = 1μm (Fig. 7(a)), with the hole radii kept constant at 100nm. Third, to create the Gaussian mirror, the beam widths are quadratically tapered from wstart = 1μm to wend = 0.7μm, which produces the maximum mirror strength (band diagrams shown in Fig. 7(a)). This procedure involves calculating the mirror strength for several beam widths (Fig. 7(b)), each takes one or two minutes on a laptop computer. As shown in Fig. 7(c), the mirror strengths are linearized after the quadratic tapering. In order to achieve a radiation-Q limited cavity, 10 additional mirror segments are placed at both ends of the Gaussian mirror that has beam width wend = 0.7μm.

Similar in the dielectric-mode cavity cases, Hzodd and Hzeven air-mode cavities can be formed by placing the
air and dielectric in the central symmetric plane of the cavity, respectively.
Again, we will focus on Hzodd, air-mode cavities and the conclusions will be
valid to the Hzeven cavities as well. Figure 8(a) shows the total Q of
nanobeam cavities that have different total number of mirror pair segments in
the Gaussian mirrors. We have achieved a record ultra-high Q of
1.4 × 109, air-mode nanobeam cavity. As shown in Fig. 8(a), the effective mode volumes of
the air-mode cavities are much larger than the dielectric-mode cavities.

Fig. 8 (a) Total Q-factors (log(10) scale) and effective mode volumes (V/(λres/nSi)3) of the nanobeam cavities for different total number of mirror pair segments in the Gaussian mirror. In each case, 10 additional mirror segments with w=0.7μm are added on both ends of the Gaussian mirror, so that the total-Q of the cavity is limited by radiation-Q. A record ultra-high Q of 1.4 × 109 is achieved with a Gaussian mirror that comprises 30 mirror segments and 10 additional mirror pairs on both ends. (b) On-resonance transmissions and total Q-factors (log(10) scale) v.s the total number of mirror pair segments in the Gaussian mirror. In this case additional mirror pairs (10 of them) are not included. A record high-T (96%) and high-Q (3.0 × 106) cavity is achieved at N = 25.

4.2. Cavity strongly coupled to the feeding waveguide

As we have pointed out, the tapering-width approach (as compared to taping hole radii) offers a
natural way of coupling the nanobeam air-mode cavity to the feeding waveguide.
Since the width of the beam is decreasing, the cavity naturally couples to the
feeding waveguide. We study T and
Qtotal dependence on the total number of mirror
pair segments in the Gaussian mirror (N) using FDTD
simulations. As shown in Fig. 8(b), we
achieve nanobeam cavity with Q = 3.0 ×
106, T = 96% at
N = 25.

4.3. Higher order modes of the air-mode cavity

The ultra-high Q cavity that we were able to design is the fundamental mode of the cavity. Higher order modes coexist with the fundamental modes inside the band gap. Fig. 9(a) shows the transmission spectrum of a waveguide-coupled air-mode nanobeam cavity, that has 15 mirror pair segments in the Gaussian mirror. The band-edge modes are observed at wavelengths longer than 1.6μm. The modes in the range of 1.2μm to 1.35μm are formed by the higher order band modes in Fig. 7(a). Figures 9(b)–9(c) show the major field-component distribution (Ey) of the two cavity modes inside the bandgap. The total Q-factors of these two modes are 23,935 and 5,525 respectively. The effective mode volumes are 2.32 and 3.01 respectively.

Fig. 9 (a) Transmission spectrum of the cavity from FDTD simulation. (b)&(c) The Ey field distribution in the middle plain of the nanobeam cavity. Resonances and symmetries of the modes are indicated in the plot. Symmetry plane is indicated by the dashed line. Length unit in (b)&(c) is μm.

5. Conclusion

We have presented a detailed analysis and a deterministic design of the ultra-high
Q photonic crystal nanobeam cavities. With this method,
Q > 109 radiation-limited cavity, and
Q > 107, T >
95% waveguide-coupled cavity are deterministically designed. These
Q-factors are comparable with those found in whispering gallery
mode (WGM) cavities [45

]. Meanwhile, the mode volumes are typically
two or three orders of magnitude smaller than WGM ones. Furthermore, energy maximum
can be localized in either the dielectric region or air region with this method.
Although we demonstrate designs for TE-like, transversely symmetric cavity modes,
the design method is universal, and can be applied to realize nanobeam cavities that
support TM-polarized modes, as well as line-defect 2D photonic crystal cavities. We
believe that the proposed method will greatly ease the processes of high
Q nanobeam cavity design, and thus enable both fundamental
studies in strong light and matter interactions, and practical applications in novel
light sources, functional optical components (filters, delay lines, sensors) and
densely integrated photonic circuits.

Acknowledgments

We acknowledge numerous fruitful discussions with M.W. MuCutcheon and P. B. Deotare. This work is supported by
NSF Grant No.
ECCS-0701417 and NSF CAREER grant.

References and links

1.

Quality factor is defined as Q=ω0EnergystoredPowerloss, and mode volume is defined as V = ∫ dVε|E|2/[ε|E|2]max.

M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16, 19136–19145 (2008). [CrossRef]

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